On the number of CP factorizations of a completely positive matrix

A square matrix A is completely positive if A = BBT, where B is a (not necessarily square) nonnegative matrix. In general, a completely positive matrix may have many, even infinitely many, such CP factorizations. But in some cases a unique CP factorization exists. We prove a simple necessary and sufficient condition for a completely positive matrix whose graph is triangle free to have a unique CP factorization. This implies uniqueness of the CP factorization for some other matrices on the boundary of the cone CPn of n x n completely positive matrices. We also describe the minimal face of CPn containing a completely positive A. If A has a unique CP factorization, this face is polyhedral.

Automated summary

A square matrix A is completely positive if it can be expressed as A = BBT, where B is a nonnegative matrix that may not necessarily be square. While a completely positive matrix can have multiple CP factorizations, there are cases where a unique CP factorization exists. A simple necessary and sufficient condition has been proven for a completely positive matrix with a triangle-free graph to have a unique CP factorization. This also implies uniqueness of the CP factorization for certain matrices on the boundary of the cone of completely positive matrices.